# Full text of "Mathematical tracts; part 1. On Laplace's coefficients, the figure of the earth, the motion of a rigid body about its center of gravity, and precession and nutation"

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(Stack >nnex IQB O'BRIEN'S MATHEMATICAL TRACTS, PART I. y^s v&)C;".X£ .£L> I 1. - MATHEMATICAL TRACTS, PART I. MATHEMATICAL TRACTS, PART I. LAPLACE'S COEFFICIENTS, THE FIGURE OF THE EARTH, THE MOTION OF A RIGID BODY ABOUT ITS CENTER OF GRAVITY, AND PRECESSION AND NUTATION. MATTHEW O'BRIEN, B.A., •MATHEMATICAL LECTURER OP CAIUS COLLEGE. CAMBRIDGE: PRINTED AT THE UNIVERSITY PRESS, FOR J. & J. J. DEIGHTON, TRINITY STREET; AND JOHN W. PARKER, LONDON. M.DCCC.XL. Stack Annex 233 PREFACE. THE subjects treated of in the following Tracts are, Laplace's Coefficients; the Investigation of the Figure of the Earth on the Hypothesis of its Original Fluidity ; the Equations of Motion of a Rigid Body about its Center of Gravity; and the Application of these Equations to the case of the Earth. The first of these subjects should be familiar to every Mathematical Student, both for its own sake, and also on account of the many branches of Physical Science to which it is applicable. The second sub- ject is extremely interesting as a physical theory, bearing upon the original state of the Earth and of the planetary bodies; it is also well worthy of attention on account of the important and exten- sive observations which have been made in order to verify it. The Author has put both these sub- jects together, commencing with the Figure of the Earth, and introducing Laplace's Coefficients when occasion required them; this being perhaps the best VI PREFACE. and simplest way of exhibiting the nature and use of these coefficients. The Author has treated some parts of these sub- jects differently from the manner in which they are usually treated, and he hopes that by so doing he has avoided some intricate reasoning and trou- blesome calculation, and made the whole more accessible to students of moderate mathematical at- tainments than it has hitherto been. In calculating the attractions of the Earth on any particle, he has arrived at the correct results, without considering diverging series as inadmissible ; and this he conceives to be important, because there is evidently no good reason why a diverging series should not be as good a symbolical representative of a quantity as a converging series ; or why there should be any occasion to enquire whether a series is di- verging or converging, as long as we do not want to calculate its arithmetical value or determine its sign. Instances, it is true, have been brought for- wprd by Poisson* in which the use of diverging series appears to lead to error; but if the reason- ing employed in Chapter in. of these Tracts be not incorrect, this error is due to quite a different cause ; • See Bowditch's Laplace. Vol. H. p. 167. PREFACE. Til as will be immediately perceived on referring to Ar- ticles 33, 34, 35, and 37. The Author has deduced the equations of motion of a rigid body about its center of gravity by a method which he hopes will be found, less objec- tionable than that in which the composition and re- solution of angular velocities are employed, and less complex than that given by Laplace and Poisson ; he has also endeavoured to simplify the application of these equations to the case of the Earth. In the First Part of these Tracts he has confined himself to the most prominent and important parts of each subject. In the Second Part, which will shortly be published, he intends, among other- things, to give some account of the controversies which Laplace's Co- efficients have given rise to ; to investigate more fully the nature and properties of these functions ; to give instances of their use in various problems ; for this purpose to explain the mathematical theory of Elec- tricity ; to consider more particularly the Equations of motion of a rigid body about its center of gra- vity, and the conclusions that may be drawn from them ; to give the theory of Jupiter's Satellites, and of Librations of the Moon ; 'and to say something on the subject of Tides. Vlll PREFACE. The Author has not given the investigation of the effect of the Earth's Oblateness on the motions of the Moon, but he has endeavoured to prove that this effect does not afford any additional evidence of the Earth's original Fluidity beyond that which may be obtained from the Figure of the Earth, and Law of Gravity. MATHEMATICAL TRACTS, PART I. CHAPTER I. FIGURE OF THE EAUTH. 1. IT has been well ascertained, by extensive and accurate geodetical measurements, that the general figure of the Earth is that of an oblate surface of revolution, de- scribed about the axis of diurnal rotation : and this fact suggests the idea, that the diurnal rotation may be in some way or other the cause of this peculiar figure, especially if we consider that the Sun and planets, which all rotate like the Earth, appear also to have the same sort of oblate form of revolution about their axes of rotation. The most obvious and natural way of accounting for the influence thus apparently exerted on the figures of the planetary bodies by their rotation, is to suppose that they may once have been in a state of fluidity ; for, con- ceive a fluid gravitating mass to be gradually put into a state of rotation round a fixed axis : it is evident that be- fore the motion commenced it would, according to a well- known hydrostatical law, be arranged all through in con- centric spherical strata of equal density ; but on the motion of rotation commencing a centrifugal force would arise, which would be greater at greater distances from the axis, and would therefore evidently produce an oblateness in the forms of the strata, leaving them still symmetrical with respect to the axis. Thus the hypothesis of the original fluidity of the bodies of our system, considered in connection with their rotation, accounts for their oblate form. 1 2 2. To account for the present solidity of the surface of our own planet, we may suppose that its temperature was originally so great as to keep it in a state of fusion, and that this was the cause of its fluidity ; but that, in the course of ages, it, at least its surface, has cooled down and hardened into its present consistence. This supposition is borne out by geological facts ; and it is by no means un- likely, if we consider that the principal body of our system is at present most probably in a state of fusion. 3. This hypothesis of the Earth's original fluidity re- ceives much confirmation from observations on the intensity and direction of the force of gravity ; for it follows from the hydrostatical law already alluded to, that the Earth, if fluid, ought to consist entirely of equidense strata of the same sort of form as the exterior surface*, and therefore the whole mass ought to be arranged symmetrically with respect to the axis of rotation, and nearly so with respect to the centre of that axis. Hence, the force of gravity, which is the resultant of the Earth's attraction and the centrifugal force, ought to be the same at all places in the same latitude, and nearly the same at all places in the same meridian. Moreover it follows from another hydrostatical law, that the direction of this force of gravity ought to be every where perpendicular to the surface. Now all this has been proved to be the case by nume- rous observations with pendulums, plumb-lines, levels, &c. (omitting very small variations, which may be easily ac- counted for in most cases). Hence, the hypothesis of the Earth's original fluidity is confirmed by the observations which have been made on the force of gravity. 4. But this hypothesis has been advanced almost to a moral certainty, by investigating precisely what effect it ought to have, if true, on the arrangement of the Earth's * We suppose the Earth to be heterogeneous, because the pressure of the superincumbent mass must condense the central parts more than the superh'cial ; besides, the well-known fact of the mean density of the whole Earth being greater than the density of the superficial parts, proves that the Earth is not homo- geneous. mass, and by comparing the result with observation; for it is found that if the hypothesis be true, the strata which com- pose the Earth ought to have not only an oblate form, but one very peculiar kind of oblate form ; and it is found that this result admits of most satisfactory comparison with ac- curate and varied observation, and actually coincides with it in a most remarkable manner; from which we may con- clude, almost with certainty, that the hypothesis is correct ; for it is extremely difficult to account in any other way for so marked an agreement with observation of such a very peculiar result. 5. The object of the following pages is to give an account of this interesting investigation, and to state briefly the manner in which its result may be tested by observation. In the first place, we shall determine the law of arrange- ment of the Earth's mass, on the hypothesis of its original fluidity, by means of Laplace's powerful and beautiful Ana- lysis; and in the next place, we shall deduce such results as shall admit of immediate comparison with observation. The most important of these results are ; The expression for the length of a meridian arc corresponding to a given difference of latitude, and. The law of variation of the force of gravity at different points of the Earths surface. The other results which we shall deduce depend on certain assumptions respecting the law of density of the Earth, and are therefore not so important. We now proceed, in the first place, to determine the law of arrangement of the Earth's mass, as follows. 6. A heterogeneous jluid mass composed of par- ticles which attract each other inversely as the square of the distance rotates uniformly in relative equilibrium* round a fixed axis : to determine the law of its arrange- ment. Take the axis of rotation as that of #, and let xyz be the co-ordinates of any particle $m, XYZ the resolved * By relative equilibrium we mean that the particles of the mass, though actually moving, are at rest relatively to each other. attractions ol' the mass on $w, p and p the density and pressure at the point (#ysf), and o> the angular velocity of the mass. Then, by the principles of Hydrostatics, we have dp = p \Xdx + Ydy + Zdz + w8 (xdx + ydy)} ... (l) To calculate the expression (Xdcc + Ydy + Zdz), let $m be any attracting particle, and x'y'% its co-ordinates ; then we have and similar expressions for Y and Z. Now assume (F) to denote the expression VV- *)2 + (y- y?+ (*' - *)2 ' i. e. the sum of each particle divided by its distance from §m. Then it is evident that y dV dV dV = Tx* ~~dy* ~dx' and Xdfc + Ydy + Zdz = —— dx + —:—dy -\ -- dx. dx dy ' dss and therefore the equation (1) becomes The coefficient of p here is a complete differential ; hence by the principles of Hydrostatics, the necessary and sufficient conditions of equilibrium are, that the whole mass be arranged in strata of equal density, the general equation to any one of them being C being a constant different for different strata, the exterior surface being one of these strata, since it is a free surface. 7- Hence the equations from which the problem is to be solved are (A). = 8. These equations are unfortunately very much in- volved in each other, so much so as to be scarcely manage- able ; for V must be found by integration between limits which depend on the form of the exterior stratum, and therefore on the equation (A) ; and also the law of density, and therefore the form of the internal strata, and therefore the equation (A), must be known in order to calculate F. But V is itself involved in (A), hence (A) cannot be made use of in calculating V. It will therefore be neces- sary to devise some way of eliminating F, without knowing what function it is. To do this in the general case is be- yond the present powers of analysis ; but in the particular case we are concerned with, the fact of the strata being all nearly spherical, introduces considerable simplification, and by using the ingenious analysis due to Laplace, we shall be able to eliminate V with comparative ease, at least, ap- proximately, but with quite sufficient accuracy. 9. In the first place, the strata being nearly spherical, we shall find it convenient to make use of polar instead of rectangular co-ordinates, and we shall accordingly transform our equations as follows : Let r, 0, (f>, /, 0', 0', be the co-ordinates of $m and $m' respectively; r, 0, 0, signifying the same as in Hymers1 Geometry of three dimensions, page (77). Then we have off = r sin 9 cos d), y = r sin 0 sin <^>, % = r cos 9, and similar expressions for v't y', %'. Hence the equation (A) becomes C = V + — r- sin2 9, 2 and the equation (/?) becomes, observing that $m = p r'2 sin 9' dr d& dfi c^ r Jo Jo \Xr'-2rr'|cos#cos#'+sin0sin0'cos(0-^)')}-fr'2 0 and r, being the limits of r', 0 and TT of 9', and 0 and 2?r of 0', 7*1 being the value of r at the surface, and there- fore in general a function of 0' and <p'. It will of course on this account be necessary to integrate first with respect to /, but it is no matter in what order we perform the integrations relative to 0' and <p', since their limits are constants. If these integrations could be performed, V would come out a function of r, 0, and <^>, and unknown constants de- pending on the form of the strata and the law of density. 10. We shall find it convenient to put /* and /u? for cos 6 and cos 9' respectively, this will give sin 9'd9' = - dp.', and the limits of /u. will be — 1 and 1 ; or we may put dfi instead of — d//, if at the same time we reverse the limits of fji. Hence our equations become (A'), When for brevity we have put cos 9 cos ff + sin 9 sin 9' cos (0 - <^>') = p, i. c. fifj! + \/l -fj? . \A - //* • cos (^> ~ 0') = P- 11. We shall now introduce into these equations the condition that the strata are nearly spherical. If the strata were actually spherical, the whole mass would be symmetri- cal with respect to the centre, and therefore V being the sum of each particle divided by its distance from $m would depend simply on the distance of $m from the centre, and therefore be the same at all points of the same stratum. We may hence conclude, that if the strata instead of being actually spherical be only nearly so, V also, though not actually the same, will yet be nearly the same at all points of the same stratum. Now the value of V for any stratum is given by the equation {A') i. e. but V (as we have shewn) ought to be nearly constant at all points of this stratum, hence the variable part of it, viz. — r2(l — /u2) must be always small : therefore since r2(l — fj?) is not always small, wz must be so. We shall accordingly take ft>2 as the standard small quantity in our approxima- tions, neglecting its square and higher powers in the first approximation. 12. Now to8 being a small quantity, we may suppose the equation to any nearly spherical surface, and therefore to any of the strata, to be put in the form where a is the radius of any sphere which nearly coincides with the stratum (that sphere, suppose, which includes the same volume as the stratum*), and aa?u is the small quan- tity to be added to a to make it equal to r, and therefore u in general will be some function of /u. and 0. Moreover, u will be a function of a also, otherwise the strata would be all similar surfaces, which of course we have no right to assume them to be ; a may be considered as the variable * We make this supposition, at present,"for the sake of giving a definite idea of what a is ; hereafter it will be found an advantageous supposition. 8 parameter of the system of surfaces which the strata con- stitute. We shall introduce the variable a into our' equa- tions instead of r, and get every thing in terms of a, /x and <£, instead of r, /u, and 0; the advantages of this change will soon be perceived. 13. First, then, in the equation (^'), putting a (l+a>2w) instead, of r, and neglecting the squares and higher powers of <*r, we find Next we shall make a similar substitution in the equa- tion (B') by putting r' = a (l + o>2w'), when u denotes what u becomes when a, /*, and 0 are exchanged for a', /, and <p' respectively. Assume for brevity, then in the equation (B') we shall have * = /oai p/ (r, /) . da', where Oj is the parameter of the exterior surface. and substituting for r,f(r,r')=tf{r,(a'+a'u?u)} dr 9 d(a'u) and hence, since — 7 = 1 + to . — - — H^ , we have da da 9 Hence, neglecting the squares and higher powers of ft)2, the equation (B7) becomes 14. This expression for V is much more manageable than that in the equation (B1) ; for in the first place the limits are all constant, and therefore we may take them in any order we please; and in the second place, p, instead of depending on all the variables, as of course it did before, is now a function of one variable only, viz. a \ for each stratum being equidense throughout, it is evident that p is the same at every point of the same stratum, and therefore varies only when we pass from stratum to stratum, i. e. it varies with a alone. Hence, changing the order of the integrations, and bringing p outside the integral signs relative to ^ and 0', we have ~ *r a'u' d* <*' da'' 15. We have thus introduced into our equations the condition of the strata being nearly spherical. We shall find it convenient to make a farther substitu- tion in this equation, viz. by putting r—a(l + <o2w), which, neglecting the squares and higher powers of <o2, gives r P {r V-! /(a' a/> dfi' w — ^ f /(a, a') dp.' d<b' aaJ<\ J -\" 'an (a' ir f f(a* a'^u> dfjL d<p} \ da'' 10 16. The next thing we shall do is to perform the in- tegrations relative to /*' and 0' ; to do this we shall expand the quantity^y^aa'), which, since it represents a'z \/a/2-2aa'p+a2' a a may be expanded in a series of powers either of — or — . a a The coefficients will evidently be the same whether we ex- pand it in powers of —. or of — , they will in fact be the a a coefficients of the powers of h in the expansion of the quantity 1 We shall assume Qy, Qn Q2, &c. to denote these co- efficients, i. e. we shall assume Q0 will evidently be unity. The rest of these coefficients will be rational and integral functions of p, i.e. of fifi + V 1 — yU2 . V 1 — //2 . COS (0 — 0'). It is evident that they all become unity when p becomes unity ; for then becomes , or 1 + h + h2 + &c. We shall have no occasion however to determine their forms. They (and other functions of the same character) are the celebrated coefficients of Laplace ; they possess very remarkable properties, which we shall now digress to inves- tigate, as they wonderfully facilitate the integrations we have to perform, and enable us to eliminate V from the equations (A") and (5"'), with great facility, without know- ing its form. CHAPTER II. LAPLACE S COEFFICIENTS. 17. IN order to investigate the properties of the functions Q,, Qn Q2, SEC. introduced in the last chapter, we shall recur to the expression from which they were origin- ally derived, viz. 1 We meet with this expression constantly in physical problems, especially those in which attractions are con- cerned, and it is therefore worthy of particular consi- deration. Assuming R to denote this expression, we have R '- x? + (yf- yf + (x'-)*' and differentiating this equation twice relatively to xyx respectively, dR __ (x-.v) dx ~ K*'-*)2+(y'-y)2 + (*'-*)2}* -JP (*'-*) « .JP'V-.)-* dx* dx and similarly fp Tt — 12 \ hence evidently d-JR d*R d*R _ 1 * =0 (1). 18. We shall express this differential equation in terms of the polar co-ordinates r0(p instead of xyx. To facilitate the transformation we shall assume an auxiliary quantity s, such that s = r sin 0 ; and therefore since x = r sin 0 cos 0, and y = r sin 6 sin 0, we shall have X = S COS 0, y = s sin 0. Then considering s and 0 as independent variables instead of x and y, we have dR _ dR dx dR dy ds dx ds dy ds dR dR . = — — cos 0 + — — sm0 (2). dx dy dzR d*R dzR d?R and — — = — — cos^0+2 • — — cos 0 sin 0+ — — sm20...(3). ds* dx2 dxdy dy2 . dR dR dR and -**,= - — * sin d> H s cosd> (4). (fm dx dy d*R d*R . d*R d?R -r = -T-r*f Sin 0 — 2 — — . S^COS 0 Sin 0 H S2 COSZ(b d0 dx2 dxdy dy2 dR ' dR Equations (3) and (5) give d?R l d*R d?R tfR 1 (dR dR ~TT + 1 T3T = T~T + ~T~9 --- — cos 0 + — — sin 0 ds* s2 d02 dx2 dy2 s \dx dy d*R d?R 1 dR by equation (2). 13 Now we have x = r cos 9 s — r sin 0*, and these equations connect % s r 9 in exactly the same way in which xys(f> are connected by the equations X = S COS 0 y = s sin 0. Hence we may prove exactly as before, that dR l (PR _ d'R d*R _ i dR d^Jr^i~d¥= "d**" + rf*2 ~ r~dr'" adding this to the equation (6), we have by (l), I. tfR rffl l^ d~R l dR l dB_ ? d0* +~d7+ r2 drF" « ds r~d7" Now by (2) and (4), dR dR cos0 dR — sin 0 + — —2- = — - , ds d(f) s dy and hence observing as before, that x s r 9 are connected together in exactly the same way as x y s 0, we have dR . dR cos 9 dR hence, substituting this value of — — in equation (7), and ds putting r sin 9 for s, and multiplying by r2, we have d*Jl cos 9 d_R_ I d^R. ,d*R dR_ _ J¥ + sin 9 ~d9 + shT^ dtf + r ~d7 + 'T ~dr ~ °' l d dR\ l d*R d f dR which is the equation (l) expressed in polar co-ordinates. * The author finds that he has been anticipated in making this use of the auxiliary quantity s, by the Cambridge Mathematical Journal. 14 19. Now K expressed in polar co-ordinates becomes 1 Vr - which = — r / r r* V 1 - %P - + -72 Qn n + &C. (See Art. 16). Hence substituting this value of R in the equation just obtained, and putting the coefficient of r" equal to zero, since r is indeterminate, we find imme- diately sin 0 dO \ d9 ] sin* 0 d(f> which is a partial differential equation of the second order, connecting Qn with /a. and 0 ; of course, being such, it admits of an infinite number of solutions besides Qn. We shall have no occasion to solve it, but "we shall find it of use in investigating the properties of Qn. All solutions of it which are rational and integral functions of cos 0, sin 0, cos <f), sin <p (i. e. of /x, \/l-fj?9 cos (f), sin 0), are called Laplace's coefficients of the nth order, having been first brought into notice by Laplace in his Mecanique Celeste, Liv. in.: the equation itself may be called Laplace's equation. Why we restrict Laplace's coefficients to be rational and integral functions of /u> 'X/l - M8> cos 0 a°d sin (f>, will appear presently. 20. We may remark here that in consequence of the linearity of Laplace's equation, the sum of any number of 15 Laplace's coefficients of the wth order is also a Laplace's co- efficient of the wth order. Also any constant quantity is a Laplace's coefficient of the order 0, for if F0 be a Laplace's coefficient of the order 0, we have d which equation is evidently satisfied by F0 = any constant, and hence any constant is a Laplace's coefficient of the order 0. It may easily be seen by trial that a/i, and a \/l - /u2 cos (0 + /3) are Laplace's coefficients of the order 1, a and /3 being constants ; and a (1 - M2)> a jtA \A ~ M2 cos (0 + /3), a (l - fj?) cos (2 0 + /3), are Laplace's coefficient of the order 2, and so on. We shall not have any occasion at present to determine the general expression for a Laplace's coefficient of the wth order, but, to give clear ideas, we shall just state that it may be put in the form A0 Mn+4} (I -fjfyM^costy + aj -I- 4,(1 -ffiMn_ 2cos(2<p + a2) , &c ............. +A(l- When AQ Al SEC. ... a^ a2 ... &c. are any constants, Mn Mn_^ &c. contain rational and integral functions of /u, of the di- mensions w, n — l, n -*2, &c. respectively*. 21. The first property we shall prove of Laplace's coefficients is this : If Ym and Zn be any two Laplace's coefficients of the mth and wth orders respectively, then i I Ym Zndju. d(f) = 0, except when m = n. Jo J -i * We shall recur to this subject in Part n. of these Tracts. 16 For since ZM satisfies Laplace's equation, we have n (n + 1) *" Ym Zn Now integrating by parts, and observing that 1 - M2 = 0, at each limit we have and similarly, f"v fz-j r~ dr.dz J, Y"^d* = - J. Tp-jf**. 1 /^ * observing that Ym -•--* is the same at each limit, be- ct(p cause Ym and Zn are functions of sin 0 and cos 0, and not of 0 simply ; hence substituting in (l) n.(n + l) r*r J0 J-\ r^ ri (/ 2,dr» -I. L(i-^^' In exactly the same way we may shew that f' -I. * This is the reason why we have assumed Laplace's coefficients to be functions of sin <f> and cos <7>, and not of <j> simply. 17 Hence subtracting \n(n + 1) -m(m+ 1) } * Ym Zndnd<t>= 0 . Now the factor n(n + l) — m(m + 1) does not = 0, except when m = n ; hence the other factor must be zero, hence r* r Ymznd/uid(f>=o, J0 J -} except when m = n. 22. Since Q0 = l (see Art. 16), we have f J — = 0, when n is greater than 0, = r^ f '0 = 47T. It need scarcely be remarked that Q0, Q15 Q2, &c. possess exactly the same properties with respect to p and 0' that they do with respect to /j. and (p. 23. We shall now have occasion to introduce a remarkable discontinuous function, but before we do so we shall give a simple example of functions of this description, in order to render our reasoning more satis- factory to those who have not been accustomed to them. We may easily prove, by the aid of the exponential value of the cosine of an arc, that *(.-f) cos a + cos (a + /3) + cos (a + 2/3) ad inf. ... = — ; suppose here that a = — , then we find 0 cos a + cos 3 a + cos 5 a + ... = - 2 sin a 18 hence this series is always zero, except when a= any mul- tiple of TT, in which case sin a becomes zero, and therefore the series becomes - ; thus though each term of this series varies continuously with a, the series itself varies discon- tinuously, being constantly zero, except when a passes through any of the values 0, ± TT, ± 2 ?r, &c.. when the series suddenly becomes -; i.e., some unknown or indeterminate quantity. To explain the nature of this series more clearly, we observe that sin — | cos a 4- cos (a + /5) + SEC. } = - ^ sin f a ) , whatever be the values of a and /3 ; suppose a = — ; then sin I a 1 becomes zero, whatever be the value of a ; hence sin a {cos a + cos 3a + cos 5a + &c. } = 0, for all values of a. Now as long as a is not a multiple of TT, sin a will not be zero, and therefore cos a + cos 3 a + ... &c. must be zero ; but if a be any multiple of TT, then sin a will be zero, and our equation will be satisfied quite independent- ly of its other factor, and hence will give us no information as to the value of that factor; hence when a is any multiple of T, cos a + cos 3 a + &c. is some unknown or indeterminate quantity. It is important to remark that the change in the value of cos a + cos 3a + &c., when a becomes a multiple of -a-, is perfectly sudden, for since the second member of the equa- tion is always absolutely zero, it is evident that as long as sin a is not actually zero, though it differs from it by ever so 19 small a quantity, cos a + cos 3 a + &c ---- must be so; for this reason cos a + cos 3 a + Sic. is called a discontinuous function. 24. We shall now bring forward the remarkable dis- continuous function we alluded to: it is the following series, viz. Qo+ 3Q/+ &c- + (2w + 1) Q«+ &c. ad inf. this series is of exactly the same nature as that we have just considered, being a function of the variables /x and <£», which is always zero, except for certain particular values of these variables. To shew this, we have Q + Q,A + ... Qnh» + &c. = , ~ , VI - 2ph + h? and differentiating this relatively to /*and multiplying by 2 h, and adding these equations, «,+ »(t*...(g. + .) «.*•+ &c. - (1--^¥ now here put h — 1 and we find 0 O, + 3 Q + &c. = - — — = , (2 - 2p)f hence QQ+ 3Q7 + &c. = 0 for all values of />, except p 0 when it becomes - : now 0 P=fJLfl' + \ -/(Z2 ^l - /li'2 COS ( - '), but cos (0 - <£') is not greater than 1 ; 20 hence, 1 -/*/*' or, squaring and reducing, (fj. — p)* is not > 0, which cannot be unless /u. = //, and this will give cos (<p - <£') = 1 ; and therefore 0 - 0' is zero, or some multiple of STT; hence the series Q0 + 3 Qx + &c. is a discontinuous function, being always zero, except when M = M' and <f> ~ <p' = 2wrr, (m being any integer,) in which case it becomes -. It is evi- dent, as in the former case, that this series is perfectly discontinuous, being zero for all values of p that differ even in the least degree from unity, and then when p = 1, sud- denly assuming the form - . 25. Now, wherever we have occasion to use the series Q0 + 3Ql + Sec., it will occur under integral signs relative to fi and 0, and the limits of <f) will be 0 and STT; hence, by what we have proved in the note* respecting the limits of * If x be any arbitrary quantity occurring in any investigation, its differen- tial dx may be defined to be any small increment of x, made use of with the understanding that it is to be put equal to zero at the end of the investigation ; and if f(x) be any function of at, its differential df(x) may be defined to be the corresponding increment of /(#), that is, The symbol j written before a differential, is generally taken to denote the quantity from which the differential is derived, that is, /«*/(*) =/(*), and the notation /**<*/(*) is taken to denote the difference f(x3)-f(xl): but this notation has a much more important signification : for in the equation put for x successively the values .r,, Xi + dn, z1 + 2dn, &c., #, + (»- \)dx, and add the results, and we find ! + dx) + d/(-r, + 2dx) &c. + df{Xl + (n - 1) dx\ 21 integrals, 0 will receive all the values between 0 and 27r, inclusive of the inferior limit, and exclusive of the superior, and will therefore never actually be equal to, or exceed 2?r. Also in all cases we shall be concerned with the same may be supposed true of <p' ; for in all our investigations, wherever 0' recurs, the value <f)' = 2 IT, or any greater quan- tity, will be only a repetition of <f>' = 0, or some value between 0 and 2 TT ; hence we may consider that <f> ~ <f> never actually equals or exceeds 2-rr; and hence it will be only for one value of 0 ~ 0', viz. 0, that p will become unity ; hence, by what we have proved, the series Q0 + 3Q, + &c. in all cases we shall be concerned with, will be absolutely zero for all values of fj. and 0, except the single values [A. = p and 0 = 0'. We now proceed to prove some re- markable properties of this series, which result from its discontinuous nature. 26. From Art. 22 it appears immediately that = 4<7T. Now, here the quantity under the integral signs is, as we have proved, always zero, except when p. = /m' and 0 = 0'; it is therefore no matter what the limits of the integration be, provided they include between them the or supposing >rj + ndx = ,r2, df(xt) + df(xl + dx)+&c. till we come to df(xa-dx)=f(xs)-f(xl) = £«<*/(*). Hence it appears that if*8 df(x) denotes the sum of a series of values of df(x), got by giving x all its values between the limits xt and xz inclusive of the former limit and exclusive of the latter; that is to say, all the values of x which form an arithmetic series whose common difference is dxy commencing with xl and ending with xz- dx. The remark respecting the limits is im- portant whenever discontinuous functions are concerned, as in our present in- vestigation ; and we must remember that though the last value of x approaches indefinitely near to x2, it never actually becomes equal to it. alues (fji = /u') and (0 = 0'), respectively ; hence, if /*,, ^2, i» 02*5 be any limits which do this, we have * (Qn + 3Q! + &c. ...)dnd<b = ±Tr. 27- In the same manner, if F (n<p) be any function of /u and 0, which is always finite between the limits - 1 and 1, 0 and Zir, F(yu</>)(Q0 + 3Q1 + &c.) will be always zero, except when JM = // and 0 = 0', and we shall have, as before, /^2 rM •^l »^l 28. Now let F(n"(f>") be the greatest value of F(^0), between the limits /ui, yua» 0i> 02 '•> ano< let ^'(/*//0//) be the least ; then it is evident from the nature of an integral, considered as a sum, that /-</>* rn* F ^/0/j ^QQ + 3 Q! + &c) d^d0 is not greater than .F(/u"0") y T 2 (Q0 + 3Q, + ...) dyurf0, and not less than i. e. (by Art. 22), not greater than 4 ?r F (^"0"), and not less than 4 TT T'1 (^^0^) ; and this is true, no matter how close together the limits * Of course these limits are supposed to be included between - 1 and + 1, 0 and 2ir. /"u Us* <£n ^a? be taken, provided /UL' and (j)' be included be- tween them. Now /*", 0", and /u//5 0//s are also always included between these limits ; hence, since /&'<£', M"> </>"» yu//s 0/x, are respectively always included between limits which we may take as close together as we please, it is evident that we may suppose //', 0", and //„» 0y/» to differ from yu'0' respectively by as small quantities as we please ; and therefore, since F (/u. <p) is always finite, we may in the above inequalities suppose F(/JL'> <^>") and F (p.^ <£„) as nearly equal to F(fj!<p') as we please, which evidently cannot be, unless FT jfr jfr 29. We shall give another demonstration of this remarkable result. Assume, as we evidently may, (Q0+3Q1+ ... Then, as before, r* T fc/Q •/ — 1 multiply this equation by d// d^)', and integrate between the same limits, and we have* It is necessary to take the same limits, otherwise in the integral /i and ^>, the values of the variables for which Q0+3Qi + ... becomes «, will not be always included between the limits, and therefore Art. 2fi will not apply to it, and our proof will be incorrect. 24 Hence, by Art. 26', differentiating this equation relatively to <p., and yu2 suc- cessively*, we have 4-TT F (fJL2(f)o) =,/*(Al2</>2); hence, since ^2 and 02 are arbitrary ,y= 47T-F; and therefore (Q0 + 3Q,+ ...) dfjidQ 30. We may hence find the value of TV' YnQndndd). Jn J -i Yn being any Laplace's coefficient ; for Yn being a rational and integral function of /m, \/l-to>2, cos ^>, sin 0-f- will be always finite ; hence we may put Yn for F (/a. (pi) in Art. 28, and we find immediately by Art. 21, PV YnQn J0 J-i * To shew how to differentiate a definite integral with respect to its limits, let f(x) + C denote the indefinite integral off'(x), then differentiating this relative to <r2, we have and differentiating relative to and in a similar manner we may differentiate integrals relative to two or more variables. t This is the reason why we have restricted Laplace's coefficients to be rational and integral. 25 where Y ' denotes what Y becomes, when /tx.' and <f> are put for (ix and (f> in it. This is a very important result ; in fact, this, and that in Art. 21, are the properties which render Laplace1 s coefficients so very useful in integrations such as we have to perform in Art. 15. 31. The equation deduced in Art. 28, interchanging fjL and (f)' for /tx and <£, shews that if F(fj.(j)) be any func- tion of fjL and (p, which is always finite, it may be expanded in a series of Laplace's coefficients ; for by this equation F (n<p) = a series whose general term is 2n + 1 rt tr /M , , , — / I F (/JL <p ) Qn d fJL U (t) . 4>ir J o »'— i Now this quantity evidently satisfies any linear differ- ential equation relative to fj. and 0 that Qn satisfies ; there- fore it satisfies Laplace's equation of the wth order ; moreover it is a rational and integral function of /x, \/l — ti2, cos <£, sin<^>, for Qn is so, and 2W+ 1 /-8W /M r will evidently differ from QB, considered as a function of ^, \/l-/x2, cos 0, sin 0, only in having different coefficients to the powers of these quantities ; that is to say, if A' be any coefficient in Qn, then the corresponding coefficient in ') Q. dp' dip 27T will be hence the several terms of the series to which F'(/u'(p') is equivalent are rational and integral functions of /u, \/l-/r, cos 0, and sin 0, which satisfy Laplace's equation, and 4 are therefore, according to our definition, Laplace's co- efficients. 32. No function can be expanded in more than one series of Laplace's coefficients. For, if possible, let Y0 + Yl + ¥„ + ... + &c. and Z0 + Zl + Z2 + &c. be two different series of Laplace's coefficients equivalent to the same function, then, since this is the case, we have Y0 - Z0 + Y, - Zl + Y, - Z2 + &c. = 0; multiplying this equation by Qnd/i.d<p, and integrating be- tween the limits - 1, 1 ; 0, 27r, we find (by Art. 22 and 30), (Y'n -Z'a) =0, "2n + 1 hence Yn = Zn ; and therefore the series are the same, and the function can be expanded only in one series of Laplace's coefficients. 33. The conclusion we have just arrived at seems to be at variance with the fact that the quantity may be expanded in two distinct series of Laplace's co- efficients, viz. but this is only an apparent discrepancy, for the function to be developed, admits of two values, one positive and the other negative, on account of the ambiguity of the sign of the square root ; and therefore ought to admit of two developments. That the above are the two develop- ments corresponding to the two values of the square root, will follow from the following proposition : viz. 27 34. To determine the sign of the series This series being the development of the expression 1 it is evident that it can never become 0 as long as h is not infinite, but it will become infinite when 1 - 2ph + h2 = 0 ; 2 that is, when p . Now 1 + h2 is always greater than 2A, except when h « 1, in which case 1 + A2 = 2A; hence, since p, being a cosine, can never exceed unity, this equation can only be satisfied by h = 1 , and therefore p = 1 ; hence cannot become infinity unless h be unity ; hence, if h be not infinity or unity, this series can never become zero nor in- finity, and therefore can never change its sign : supposing then that h is not infinity or unity, will always have the same sign whatever be the value of p. Hence, supposing p = 1, and therefore Q0 = 1, Q1 = 1, &c. (Art. 16.) it is evident that Qo + Qih + &c. has the same sign as 1 + h + h? ... 1 i. e. as - -; 1 -h hence, if h be less than unity, this series is positive, and if h be greater than unity, it is negative. In the same way it may be proved that tilc samc 8'n as \ 7 g -r* 7 - ' rtf ft ft -~ 1 and is therefore always negative when h is less than unity, and positive when h is greater than unity. Hence, when h is less than unity, i QO + Q\h + Qzh2 + is the positive value of V 1 - 2ph + tf and - \ Qo + Qi 7 + . . . \ the negative value ; h ( h } and when h is greater than unity, the reverse is the case. 35. To represent, then, the true general development of —j- , let A; be a discontinuous function of h, Vl — %ph + h* such that A: = 1, when h is less than unity; and k = 0, when h is greater than unity. Then, whatever h be, the positive value of — ^ V ] will evidently be and the negative value will be (1 -k} {Q0+ Q,h + Q2A2+ ...} + k JQ0^ + Q ^+ ...1. Thus, in reality, , , if we restrict our- Vl-2ph + hz selves to only one of its values, can be developed in only one series of Laplace's coefficients. The conclusion we have just arrived at respecting the true development of this quantity, will be highly important in our future investi- gations. 36. We shall conclude this chapter with the following important property of Laplace's coefficients, viz. 29 If Y0+Yl + F2 + &c. = 0 (l), be any equation arranged in a series of Laplace's coeffi- cients, IJL and 0 being indeterminate, then we must have F0 =0, Y,= 0, F2 = 0, &c. for, multiplying (l) by Q,,dfj.d(p, and integrating between the limits - 1, 1, and 0, 27r, we have, (by Arts. 22 and 30,) 2tt + 1 and therefore Yn = 0. We have now considered these remarkable functions at sufficient length for our present purpose ; we shall recur to this subject in Part u. of these Tracts. CHAPTER III. FIGURE OF THE EARTH. 37. WE are now prepared to return to the equation (5'"), see Art. 15 ; the properties we have proved La- place's coefficients to possess, will enable us to perform the integrations relative to // and <£' with great facility. In Art. 16 we stated that we should expand the quan- tity f (a, a'), or / , in either of the series Now there is an ambiguity in this quantity, since on account of the square root, it admits of two values, one positive and the other negative ; but, on referring to Art. 6, it is evident that the square root is always supposed to have its positive value; for the distance between $m and $TW', which is expressed by this square root, is evidently taken to be the absolute or numerical distance, without reference to sign ; hence, by what has been proved in Art. 35, neither of the above series will give the true general development of y(«, a'), but we must put /(a, a') = *. Q0 + Q + Q2 + &c. + (1 - fc)a' JQ0 + Q,^ + Q2^+ &c.}, where k is a discontinuous function of — , which is always unity while a is less than a, and zero when a is greater than a. For brevity we shall take An to denote the co- efficient of Qn in this series ; i. e. and then, we shall have 38. What we have proved of Laplace's coefficients naturally suggests the advantage of expanding the quan- tity u in a series of Laplace's coefficients ; this may be done, since, on account of the assumption we have made respecting the nearly spherical form of the strata, u can never become infinite ; we shall assume, therefore, u = u'0 + u\ + u'2 + &c. tt'o, M'J, &c. being Laplace's coefficients of the orders 0, 1, 2, SEC. and functions of a', /u', and (£'. We shall denote by uot ult u2, &c. what w'Q, u ^ u'2, &c. ... become when a, yu, and (f> are substituted for o', /n', 0', and therefore we shall have u = u0 + ul + u2 + &c. We shall also have occasion to make use of the values of w'0, 7/n w'j>, Sec. when /u. and (f) alone are put for /u.' and 0', a' remaining unaltered, These values we shall denote by 'w0, fuiy 'w2» &c- 39. find and J9. Hence in the equation (5'"), see Art. 15, we immediately by Arts. 22 and 30, f 32 and hence, putting u0 + w, + w2, &c. for u, y _ 47r / -• y i ^ . 2 _ /.. ... . .. . o— \ a^o = 4?r / ' p A0da •/« + a series whose general term is 4-7T <y* f ' p' { aun — — + - — -. (a An'un)\da . Jo \ n da 2n + 1 do' v J 40. If we put for A0 and .4n their values, viz. a'2 an+2 a" k — + (l-k)a', and k — + (1 - A?) ^ , (see Art. 37.) ; the factor of 47rfc>2 in this general term be- comes Now /c is always unity while «' varies between the limits 0 and a, and zero while a varies between the limits a and al (a being of course never greater than a^ ; hence that part of the quantity under the integral sign which is multiplied by k will not exist except between the limits 0 and a, and that part multiplied by 1 - k will not exist except between the limits a and al ; hence this integral may evidently be put in the form -UJLf* p'a'2da'+-r ~ r (a p — (XXn+s) da a Jo ^ (2n + 1 an+l J0 r da' y t , d f 'u. \ p —, \~-. } ^ da \an-2J 33 We shall, for brevity, denote this expression by <*n "„, <rn being merely a prefix assumed to express a certain operation performed on WB. In the same way the first term of F, viz. 4-Tr Ip A0da', "0 will evidently become 4>7r Ty { k — + (l - k) a] da, or 4>Tr J p a'2 da + 4nr f l p a da . Hence the development of V becomes finally F = — fa pa2da' + 4>7r / ' pa da a Jo r Ja ( 4- 4-TTca2 cr0w0+ ff\^\ + cr2w2+ Sue. It is evident that this is a series of Laplace's coefficients, the sum of the first three terms being a coefficient of the order 0, and the succeeding terms of the order 1, 2, 3, Sec. respectively. 41. We shall now express the value of Fgot from the equation (A") (see Art. 13) in Laplace's coefficients; to do this we observe, by trial, that -^ — /m2 is a Laplace's co- efficient of the order 2 ; hence the value of V got from (A") will be arranged in Laplace's coefficients by simply putting it in the form 34 42. If we now subtract the two equations we have thus obtained, we shall eliminate V and arrive at an equa- tion consisting of a series of Laplace's coefficients, which, by (Art. 36), must be put separately equal to zero; hence we get the following equations, viz. — 'o'^d' ' l pada' + 4>Tra)2(r0u0 — C + - = 0, a Jo ' Ja ' 2 — (^ - fjf) = 0, and a-nun = 0, for all values of n except 0 and 2. Thus we have eliminated V without knowing what function it is, and obtained equations for determining w0, MJ, uZ9 &c., and therefore the equation to any stratum. 43. We shall now proceed to solve these equations, commencing with the last of them, viz. <rnun = O...(l), except n=0 or 2. By (Art. 40) this equation is equivalent to -^ rapfartda' + - - l— - - fa p-^.(a'n+*fun)da a Jo ^ (2n + l)o"+l./o ^ da a" ro, , d ( '«. \ , / P -r-7 -r-i )da = °- 1 Jo r da k«*~y + 2n + Now, by Note, p. 24, if we multiply this equation by aw+1, and differentiate it relatively to a, and then multiply it by — , and differentiate it again relatively to a, we shall by this process get rid of the two last integral signs, and arrive at a differential equation which may be put in the form • dun A +4...0 ...... (2), 35 Alt Ay being functions of a and p, which we have no oc- casion to determine. Now let v and v' be two quantities which satisfy the equations crnv — an = 1 ......... (3), and let C and C" be two constants, then un = Cv + CV will be a solution of the differential equation (2) ; for, per- forming the operation an on both sides of the equation, un = Cv + CV, we have &nun = crn(Cv + CV) = C(TnV + C'(Tnv' (evidently from the nature of the operation <rn) or ffnun = Ca" + C' —^ , by (3). Now if we get rid of the integral signs in this equation by the same process we have applied to the equation (1), the second side will evidently be made zero by the differenti- ations, and thus we shall arrive at the same differential equation as before; hence »,-Ca + CV« is a solution of (2). Now this is evidently true whatever be the values of C and C"; hence this solution contains two arbitrary constants, and is therefore the most general solution that (2) admits of. Hence all values of un which satisfy (l), since they also satisfy (2), must be values of * It is evident that v and v' are two different functions, otherwise the equations (3) would give which is not the case. This remark is necessary, for if v and t>' were not different functions then C and C' would add together, and therefore be equi- valent to only one constant. Cv + C'v. To determine what values of Cv+C'v' satisfy (1), substitute Cv + C'v for un in (1), and we find or Ca" + C' — = 0, by (3). Now this equation ought to be true for all values of a ; hence (7 = 0, and C'=0; hence it is evident that only one value of Cv + C'v, namely zero, satisfies (l); and hence it follows from the equation (l), that Thus un is zero for all values of n except 0 and 2 ; this result produces a considerable simplification in the equation to the strata. 44. We shall next consider the equation involving w2, which may be written thus, In this equation we may conceive w2, which is a function of n, vi - ^, cos 0, and sin 0, to be developed in a series of powers of i — /j? and <£. Let y (^ — yuc)m0" be any term in this development, -y being an unknown coefficient to be determined, then the corresponding term in the equation will be except m = 1 and n = 0, in which case it will be Now i — /a? and 0 are arbitrary ; hence we must put the coefficients of their several powers separately equal to zero ; and hence for all terms, except that in which m = 1 and n = 0, we have 0"27 = ° i and therefore y = 0, as before in (Art. 43) ; 37 and for the term in which m = 1 and n = 0, a2 ff*V + ^.= \ which equation will determine y ; hence we have w2 = 7 (i ~ M8)» where -y is given by the equation 45. We cannot determine UQ from the remaining equa- tion in Art. 42, on account of the. unknown constant C in- volved in it ; but the value of u0 follows from the assump- tion that we have made respecting a in (Art. 12) ; namely, that it equals the radius of the sphere of the same capacity as the stratum whose parameter it is: for this assumption gives — => volume included by stratum, 3 rtir r 1 /« I. Li a3 rz-T jjf a(l+a>8u) 47ra3a>8w0> by (Art. 22.) Hence UQ = 0. 46. Thus we have determined the values of w0, M,, w2, &c. ; and it now only remains to substitute these values in the equation, r = a . { I -i- or (MO + ut + u2 &c.) | ; 38 and we find that the equation to any stratum whose para- meter is a, is Now the equation to a spheroid generated by an ellipse revolving about its minor axis, is V 1 - e2cosa l^ - 9\ a being the major axis, e the eccentricity, and 0 ,the angle which r makes with the minor axis. Supposing e very small, this equation becomes e2 e~ r = a \l + ~sin20* 2 2 = a {l -€cos20j, e being the ellipticity, maior axis — minor axis for the ellipticity = — ? : : major axis -, nearly. This equation may evidently be made to coincide with the equation to the stratum, by putting a e = o to2 ; to2'/ and therefore e = - :: — = ft)2/Y» nearly. Hence the strata are all spheroids of revolution about the polar axis ; the ellipticity of any stratum being ufy, y being got from the equation o2 _ STT or putting in this equation — for y, the equation for deter- (t) mining the ellipticity of any stratum, will be 8?r or, by Art. 40, — / p a da' H / p' — "' (a'5e')dar a Jo 5a Jo da a8 /-"i ,de 0*0? + — p — ; da + = 0, 5 Ja ^ da' STT which equation will give e when p' is known as a function of a'. 47. Thus we have arrived at the remarkable result that the mass must be arranged in strata of equal density, which are all spheroids of revolution about the axis of rota- tion, their ellipticities being connected by the equation just obtained. It is evident that our investigation gives us no infor- mation respecting p ; hence the law of density of the strata is quite arbitrary, and must be determined, if possible, by some independent method. 48. We shall presently shew that the results of our investigation may be compared with observation in a most satisfactory manner, without knowing any thing of the law of density of the strata. This is fortunate, as we have no means of determining this law, but must have recourse to an hypothesis which it must be confessed is rather empirical ; but as the results it leads to may be made to agree well with observation, we must look on it as probable. The hypo- thesis we allude to is this, that the variations of the pressure in the interior of the earth (supposed fluid and of the same chemical constitution all through,) are proportional, not to the variations of the density as in gases of uniform tempe- rature, but to the variations of the square of the density ; i.e. that instead of having dp = kdp, we have dp = kpdp. There is some slight reason a priori for assuming this formula, for it is evident that p ought to increase more rapidly with p in the fluid composing the earth, than it would do in gases, both on account of the incompressibility of that fluid, and the increase of temperature as we go towards the centre ; and hence kp dp will represent the variations of p better than kdp. But the chief reason for assuming this formula is, that it leads to correct results, and simplifies the equations we shall be concerned with, as will appear. 49- Assuming then this connection between the pres- sure and density, we may calculate the law of density from the equations .................................... (1), V = ^-J^ p'a'da'+tTT rip'a'da ......... (2); which are got from (Art. 6) and (Art. 42) ; neglecting or, for the centrifugal force will make a very little difference in the law of density, and it will be useless to be very accurate here, as we are proceeding on rather uncertain grounds. We have, from (l), dp_ dV d^** P da multiplying this by — , and differentiating relatively to a, P we have d Substituting the assumed value of dp, viz. dp — kpdp, we have which may be put in the form Therefore, putting — = g8, pa = A sin (70 + 5), sin and p = A and B being arbitrary constants. 50. To determine B, let a = 0, then we have p = co , unless J9 = 0 ; hence since p, as we may evidently assume, is not infinite at the centre, B = 0, and we have A sin qa P ~ a To determine A and q, let pj be the density of the superficial parts of the earth, i. e. the value of p when a — a, , and let D be the mean density of the earth ; then mass of earth its volume 6 47T 42 / pa* da' JQ s 3 A 3A ra, . = — — I a sin q a da ; a3 J0 "T ^ sin go' 3A - . also 0i = — sin a, hence sin 9^! — qal cos graj 9 a\ JJ from which equation q may be found, and then we shall have A from the equation sin 9 a, 51. Observation shews that — is about — , and on D 11 substituting this value of --j- in the equation for deter- mining (7, we shall find by repeated trials (which is the only way we can solve it) that it admits of several solu- tions, of which one only leads to right results; it is this, <7«i = about — ; 1 5-7T . hence, substituting — — for (7, we have . . /57T a A sin | — — o a. 43 To determine A, we shall put a = a, in this, which gives A . 5-n- « sin T - sin 30° and hence .4 = S^e^, and we have O which, if our hypotheses be correct, expresses the density of any stratum in terms of the parameter of that stratum, and the superficial density. 52. The method by which we have arrived at this formula for the density is not very satisfactory, and we shall therefore consider it as empirical ; we observe that it gives a density which increases as we go towards the centre, but does not become infinite there; this is most probably the case; it also makes the pressure vary more rapidly as we approach the centre than it would do if the earth were gaseous and of uniform temperature; this is also most probably the case; and it gives the mean density of the earth its proper value: we shall prove presently that it gives the true value of the earth's ellipticity, and also the true value of the coefficient of precession; hence, on the whole, we may assume it with some probability as the law of density. 53. Hence, finally, it follows from the hypothesis of the earths original fluidity ; (1) That the earth ought to consist of equidense strata, all spheroids of revolution about the axis of rotation. 44 (2) That if e be the ellipticity of any of these strata it satisfies the equation - i/v«w + A/V/-- <•• vv°' a Jo r 5a?Jo r da a2 /">i ,de . , a2 ' a /">i ,e . , a <t) + — / p — , da'+ - = 0. 5 > r da STT (3) That we may assume with probability the law of density to be a /STT a\ p = 2/0, - sin — - ) . ri at \ 6 aj CHAPTER IV. METHODS OF COMPARING THE RESULTS JUST ARRIVED AT WITH OBSERVATION. 54. IN order to test the correctness of the conclusions we have just arrived at, we shall now deduce from them results of a more practical character, which shall admit of direct comparison with observation : the first result we shall deduce is this ; If * be the length of a meridian arc, measured from the pole to any place whose colatitude is c, then a and ey being the major axis and ellipticity of the earth. For, by Art. 48, the equation to any meridian is r = a (1 - e, cos2 0) = «(l--cos20) ......... (1). XT ^^ Now_= r* = r, neglecting squares &c. of e, _«{!-!.<_ | COg 20} by (,), therefore, integrating, (2). 46 Adding no constant because « evidently = 0 when 0 = 0. Now c (the colatitude) is the angle made by the normal with the polar axis ; hence (9 — c) is the angle made by the normal and radius vector, and hence or, neglecting the squares &c. of small quantities, 9-c = e sin 20, by (1), therefore 9 = c + e, sin 2 9 = c + et sin 2 (c + e sin 20) = c + e/ sin 2c nearly ; hence, substituting in (2), we have a \c + e sin 2c -- - c — - sin 2c|, 2 4 55. To shew how this result may be compared with observation, let s' and c' be the values of s and c correspond- ing to another place near the former, and on the same meridian, then and therefore *' - s = a j 1 1 - |) (c' - c) + ^ (sin 2c' - sin 2c)| , cos ''+ c> sin c'-6'' 47 Now in this equation s' — s, being the distance between two places near each other, may be determined by the usual method of triangulation ; cos (c + c) may be found by any of the ordinary methods of determining the latitude of places, without aiming at any great accuracy, since it is multiplied by the small quantity e ; c' — c, not being mul- tiplied by a small quantity, must be determined more ac- curately by observing the meridian zenith distances of the same star at the two places, and taking the difference which will evidently be equal to c — c ; thus we may put our equation in the form when A, B, C are known ' quantities got by observation. In the same way, by observations at other places, we may obtain any number of similar equations; suppose them to be &c ....... &c ....... From any two of these equations we may determine a and e, ; and the values of a and e so determined* ought to satisfy all the other equations; hence, if we find that all these equations are satisfied by the same values of a and ey, it is evident that our result agrees with observation. 56. Now a number of meridian arcs have been mea- sured, and a system of equations similar to the above have been formed, and it is found that the values a = 3.Q62.82 miles, e = — , ' 306 satisfy them all to a remarkable degree of accuracy, allowing for certain small errors which may be easily accounted for ; and which, even considering them in the most unfavourable point of view, are very much smaller than e,, which is itself * Or rather, the values of a and e, got from all the equations by the method of least squares. 48 a very small quantity ; and indeed if we bear in mind the delicacy and number of the observations requisite in order to form the above equations, the smallness of the errors is most remarkable. 57. On the whole we are justified in concluding from observation, that the equation to any meridian and there- fore to the earth's surface is very nearly this, viz. r = a { 1 -e/cos80}, where a = 3902.82 miles, and € — — . ' 306 Hence the hypothesis of the earth's original fluidity leads to a very peculiar result, which is capable of varied and extensive comparison with observation, and which agrees with it in a remarkable manner; from which we must conclude that this hypothesis is most probably true. 58. The second result we shall deduce from our theory is this ; If g be the force of gravity at any place whose colati- tude is c, then g = U [ 1 + I — 61 I COS'C^ . Where G is a constant, namely, the value of g at the equator, and m the ratio of the centrifugal force to gravity at the equator*. * To determine w, we observe that G - 32.2 + a small quantity, hence ura 32.2 + a small quantity In putting 32.2 for G we have assumed a foot to be the unit of length, and a second the unit of time; hence we must express a and to2 in terms of these units, and therefore we have a = 3962 miles, nearly, = 3962 x 5280 feet, and 49 To prove this, we observe that -g is the resultant of the forces which act on a particle at the surface, which forces are, by (6), dV dV dV — + ftT<r, — - + ory, — — , das dy dz and therefore g must balance these forces ; hence, by the principle of virtual velocities, if dr be any variation of r, and da?, dy, d#, its resolved parts along the axes of co-ordinates, and >// the angle which r makes with the direction of g, we shall have .r ) dx + ( -- \- u?y }dy + — dsr, \dy / * dx which gives I \d.T. " r Now g acts in the normal (by the principles of Hydrostatics), and the normal evidently makes a very small angle with the radius vector ; hence \|/- is very small, V and therefore, since cos \J/ = 1 - — + &c., we may, neglect- ing squares and higher powers of small quantities, put cos \js = 1 ; moreover we have, by (6) and (42), and 4-Tr r<i , ,n , , /*", , . , , w2a2 C = — / pa* da + 4-TT p a da + - ; a Jn f Ja ' 3 Htld "> = 24ho^ 2-r 24x60x60' Mnkins; the<:e substitution*, we shall find m = about — — * 50 hence, observing that a is a function of r in virtue of the equation r = «{l+e(i-M2)}, we have, substituting in (1), dC g=-dr _dCda dr dr *TT ra , , 2ft,2 a\J d(ae) \ a da — l ~ - (-M which, neglecting the product of small quantities, may evidently be put in the form where G ** -— I p'a'ada' + small quantities. Since g *= G when /u = 0, it is evident that G is the force of gravity at the equator. (Of course, in all our formulae, at is supposed to be put for a after all differentiations and integrations have been performed, since the particle on which g is the force is supposed to be at the surface.) Now by (53), -1 (ap'a'zda'+~ ^ o'-^ a Jo P 5a5 Jo r da a2 /*«i ,de , , + — / P -r-, ; da + - = 0. 5 Ja ? da STT Multiplying this equation by a3, and differentiating relative- ly to a, we have d(ea?) ra , , ra, ,de , 5ft,2 a4 - / oVW+a4 / o— ,da'+- - = 0, da Jo r J" r da STT 51 or, observing that af is to be put for a, 8?r / p a> da _ 5g,V = TcT' since G = ~ f" pa'*da + small quantities, fl ''O 5ma = 5 centrifugal force a>2o since m = ; = — - ; gravity at equator G hence — = — - e da a da 5m and hence equation (l) becomes since 0 = c + esin2c, by (64). 59- To shew how to test this result by observation, we observe that if p be the length of the seconds pendulum, then since 1 = 27T \/- , we have p = ^L x-i when P = — 2 = value of j» at the equator. 52 Now p may be determined by observation at any place ; hence, by observations at different places, we may find a system of equations such as before ; viz. See. &c. when Ay B, A', B"> SEC.... are quantities got from observation. Now it is found that the values , 5m P = 39.01228 inches, and -- e/ = .005321, satisfy all these equations, not so exactly as before, but yet with remarkable accuracy, considering the small quantities we are engaged with; hence this result is another proof of the Earth's original fluidity. 60. The comparison of this result with the former is a strong additional proof; for the former result gives and since we know that m = - , the present result gives this is a very remarkable coincidence, and must be consider- ed as a decided proof of the correctness of our hypothesis. 61 . If G' be the value of g at the pole, G'=G _ and 53 G'- G 5m / This result is Clairaut's theorem. 62. We may determine the Earth's ellipticity by means of the law of density assumed in Chap. in. The equation for determining e is - 1 f'p'aPda' + ~ f V J~, (ea*)daf aJ0 r 5asJ0 r da a /•«, ,de' , «r a* + -/ P^-7da'+ -- = 0. 5 Ju r da 8-rr Integrating the second and third terms by parts, we have a2 r^du , , , — a r^u , , , of a -- / — , e da + L- - 5 Ja da 5 5 Ja da 5 8?r A sin qa Now putting p = — — , we have / p'a'~da'= — (sin qa — qa cos qa) dp A , -1-7 = -- - (sin q a - qa cos qa ) ; hence, if we assume e (sin qa - qa cos qa) = »;, (1) will become At) A ra , t , A a* /-a, 1J. . , , O€. (O*\ , --5-+— ,/ via*dc!+— - -±da+(^+ — a2 = 0; (f- a 5aiJo 5 Ja a'* \ 5 S-rrJ dividing this by a2, and differentiating relatively to a, we have A d / ri\ A ra , , - - — [-J - - / na da '= 0 ; tfda\\arj erJ* 54 or multiplying by a6, and differentiating again, 63. To solve this differential equation we shall assume £' such a function of a that therefore Hence, substituting in the equation (2), we have a??-3a fVr'da' + (fa /V Fa'? da'* ; »/0 ''O •'O or dividing by a and differentiating, *g-«i;+ *".f *?'«•: or again dividing by a and differentiating, hence <£= C sin (qa + C'), C and C' being arbitrary constants, and r a'^da = — {sin (qa + C') -qacos(qa + C')}, and {"a faa<gda' = — {3 [sin (qa + C') - qa cos (qa + C')] 55 Hence we shall have, putting C instead of — , rj = C | -y-j [sin (qa + C')-qa cos (qa + C")] - sin (qa + C')> . 64. We might determine the constants C and C' by substituting this value of rj in (l), Art. 62, but the following method will be more simple : in the first place we may see, a priori, that C' must = 0, for otherwise we should have »/, and therefore e very large when a is very small, contrary to our assumption of the nearly spherical form of the strata ; hence rj «• CJ-jj (sinqa - qacosqa)- sin qa > ...... (2), and therefore, since rj = e (sin qa — qacosqaj, . -- q2a2 sin qa - qa cos qa) i s 2\ We shall determine C by means of the value of ^ea ' , da got in p. 51. Multiplying (2) by a2, and differentiating relatively to a, we have d(r,a2) — = Ca (sin qa — qa cos qa), da also doing the same to the equation tj = e (sin qa— 70, cos qa), we have — ; = — i (sin qa - qa cos qa) -f eg2 a3 sin qa. da da Equating these values of , we have da 1 d (ea8) sin qa '+^0*^ oa sin ga - qa cos 70 56 . d(ea3) 5 ma Hence putting a = a , and --- - = - •* , see p. 51, 5m 36/p, = TH -~D~' also putting a = nf in (3), we have by Art. 50, from which two equations we get 5m "i" »H- D XT 1 5ir p 5 rsow m = , qa = — , — = — , 289 ' 67) 11 substituting these values and reducing, we find e = about — . 306 This result agrees with observation, but the agreement is not of much value on account of the assumption of the law of density. 65. The effect which the attraction of the Earth has on the Moon's motion, is usually brought forward as an- other means whereby we may test the correctness of the hypothesis of original fluidity ; and the agreement between theory and observation in this particular is considered to afford additional evidence of the truth of the hypothesis. We shall attempt however to prove that this is not the case. To do this, we shall shew that if the equation to the Earth's surface be known, and also the law of variation of the force of gravity, then the effect of the Earth's attraction on the Moon follows as a necessary consequence, independently of any theory except that of universal gravitation. 57 66. It is evident from the smallness of the variations of the force of gravity, that the Earth must consist of nearly spherical strata; hence all the results we have al- ready obtained, so far as they depend on the nearly spheri- cal form of the strata, will be true whether the hypothesis of original fluidity be correct or not. Hence, as before in p. 49, we shall have putting a/ for r in the small term. * Now the expression for V in Art. 40. may evidently be put in the form a series whose general term is + 47TO)2 , d hence, differentiating relatively to r, and observing that a is a function of r in virtue of the equation r = a (l + o>2w), and supposing that at is put for a after the differentiations, we have flTF__47r /.Ol , , /47r „ \ da ~a7~~^J0 pa * For the part of V which is multiplied by 4ir £p'a'eda' is evidently ,&c.)}, which =-. 58 a series whose general term is ( a series wnose general term is \ 1 (da <. n + 1 ra, d „ , , , , > , \--r - -= / P — ,(una*+3)da + pauA dr I 2n+l«n+2 Jo r da'v ' ' ' ") , + 47ro>z - (2n+l)«< Now since r = a (l + ct>2w), c?a and therefore — =1 - &> da we have, evidently, neglecting o>4 &c., MTT \ da _ f \rt ?' ' * ') dr P' ' and therefore, neglecting a>4 &c., 4£--?r>w a series whose general terra is 47TO)2 hence if, for brevity, we put shall have -o>2ai (1 -/x2); and hence 47Tft)2 , 2 3 — 5- { za + - z, + -5 z2 + &C. 5 59 Now if we suppose rj and g known, it is evident that the first member of this equation will be known, and may therefore be supposed to be expanded in a series of known Laplace's coefficients ; and hence, since the Laplace's co- efficients of different orders on each side of the equation must be separately equal, by Art. 36, the values of Z0 Zl Z2 &c. will be known. * Now it is evident immediately, from Arts. 14. and 40, that the value of V for any external point is V= — faip'a'*da + 47rte)2(- Z0 + - - Z, + 4 Z2 + &c r Jo [T r r hence, since Z0 Zt Z2 SEC. are known, as we have just proved, it is evident that the value of V for any external point is also known. 67. Hence, if we know the form of the Earth's sur- face, and the law of the variation of gravity, we shall know the value of V for any external point, and there- fore be able to determine the attractions of the Earth on that point without making use of the hypothesis of original fluidity. Hence it follows that if the form of the Earth's surface and the law of variation of the force of gravity, calculated on the hypothesis of original fluidity, agree with observa- tion, then the effect of the Earth's attraction on the motion of any external body, such as the Moon, calculated on the same hypothesis, must also agree with observation, whether that hypothesis be true or not; and hence we conclude, that the motion of the Moon does not afford additional evidence of the Earth's original fluidity. * For the only difference made in the reasoning in Art. 40, by using the expression for V given in Art. 14 instead of that given in Art. 15, will be simply this, that we shall have to consider — instead of — ; also, since the attract- ed point is external, and therefore r always greater than a', it is evident that k will be always unity. In the next chapter we shall determine the equations of motion of a rigid body round its centre of gravity, and thence deduce the Earth's motion round its centre of gravity. We shall find that the result affords a confirmation of the law of density assumed in Chapter in., and also of the hypothesis of original fluidity. CHAPTER V. EQUATIONS OF MOTION OF A RIGID BODY ROUND ITS CENTRE OF GRAVITY. 68. WE know from the principles of Dynamics, that a rigid body acted on by any forces moves relatively to its centre of gravity, in the same manner as if that point became fixed, all other dynamical circumstances remaining unaltered; hence, whenever we wish to investigate the motion of a body relatively to its centre of gravity, we may consider that point as fixed, and this will render the investigation simpler. Suppose, then, that we have a body whose centre of gravity is fixed, acted on by any forces ; let §m be any ele- ment of it, any % the co-ordinates of Sm at the time t referred to any arbitrary rectangular axes fixed in space, and origi- nating in the centre of gravity ; let L M N be the moments of the impressed forces round the axes of a? y % respectively, then we have, by the principles of Dynamics, tfco — - * - -— « U M (A). 69. In order to perform the integrations denoted by 2 in these equations, we shall introduce new variables instead of xyz, which shall have reference merely to the- position and motion of the whole body, and not to any particular particle of it. To do this, Let x y' % be the co-ordinates of $m referred to any arbitrary rectangular axes fixed in the body, then we have x = se cos (x x) + y cos (y' x) + % cos (% x) ; differentiating this relatively to #, and observing that x y %' do not vary with #, we have dot , d cos (a?' a?) ,d cos (y' x) , d cos (z'x) dt=* — dT-~ + y~— dT~ +%~~dt — ' Now the axes of x' y % are perfectly arbitrary ; we may therefore suppose them so chosen that they shall coin- cide with the axes of x y % at any instant we please. Suppose therefore that this coincidence takes place at the time #, then we shall have x' = x, y = y, % = % ; and if,. „ d cos (x'x) d cos (y'x) for brevity, we denote the values of -^ — - , - y— - ,. at at ~ — -, at the instant of coincidence by X X' X" re- dt spectively, our equation becomes dx — = \x + \y + \ #; at X X' X" are evidently variables which have reference to the position and motion of the whole body, and not to any particular particle of it, for they depend simply on the angles which the two systems of co-ordinate axes make with each other at any time, or rather upon the rate at which these angles are varying at the instant of coincidence of the axes. In the same way we may prove that dy -j~t = M + ft % + fji 'x, dss „ — - = vss + i >x 4 v y; at where /M // n", v v v" are quantities similar to X X' X". 63 Now since $m is rigidly connected with the origin, we have xz + if + s? = constant, dx dy dx and therefore x— +y— + ss— =0; dt dt dt „ dx dy dz substituting in this equation the values of — -~- — dt dt dt just found, we have Xa?2 + fiy2 + v%* + (X'-f fj.")xy+ (p +v")yz + (v + X'')#a? = 0; hence, since xyz are arbitrary, we have X' + JM" = 0 // + v" = 0 v' + \" = 0. . dx dy dss , Hence the values of -p- — — become dt dt dt - dy dx — =vy-Xx. To conform to a common and convenient notation, we shall put o>j ft>2 0)3 instead v" X" /^L" respectively, and write these equations thus, dx — .„,»-«* — - = W3ar-tt,2f at we shall presently determine what ojj w2 o>3 are. These equations express the relations which exist be- tween the velocities of any element of the body and its co- 64 ordinates at the time £, in consequence of the rigidity of the body. The substitution of these values of — — — at at at in the equations (A) will be very advantageous, since o>i c«2 ft>3 are independent of x y z, and may therefore be brought outside the integral sign 2. To perform this substitution we have from the equations (J5), differentiating dx dy dz . and m the result putting for — — — their values given dt dt dt by the equations (J5), d<o3 Now the axes of so y % are perfectly arbitrary, we may there- fore choose them so that the principal axes of the body shall coincide with them at the time £, and therefore we shall have at the time #, and hence, from the equations just obtained, we have at the time # at dt dt and hence the first of the equations (A) will become ~^m (a? + f) + w^ 2$m (* - y*) = N. Now the principal axes coincide with the axes a? y z at the time t, hence if A B C be the principal moments of inertia of the body, we shall have 65 23m (x* - y2) = (y* + *) and hence ^. — — \- (B — A) AT. We may transform the other two of the equations (A) in exactly the same manner, hence, instead of the equations (-4), we have the following three equations, in which the integral sign 2 is got rid of, viz. A — + (C - B) ft>2o>3 = L at at 70. It remains to determine what wl w2 w3 are; the equations (B) will enable us to do this immediately, for from these equations we find (putting % = 0) that the ve- locities parallel to the axes of x and y of any particle in the plane of any are daa dy -= «,,,*; and hence the whole resolved velocity in the plane of xy of that particle, viz. will be o)3 <\/a? + y2. Now if we suppose the particle to be at a distance unity from the origin, and therefore a?2 + y2 = 1, this velocity will become a>3, and hence w3 is the resolved velocity in the plane of xy of any particle situated in that plane, at a distance 9 66 unity from the origin. Since the particle is rigidly con- nected with the origin, it is evident that this velocity takes place perpendicularly to the line joining the particle and the origin ; and also since — is negative, it is evident that this dt velocity tends to move the particle from the axis of x to- wards that of y, In the same way it may be proved that MI and <o2 are similar velocities with respect to the planes of yz and zx, respectively ; o>i tending from the axis of y towards that of #, and o>2 tending from the axis of % towards that of sc. 71- To give clear ideas, we shall represent the manner in which these velocities tend, by means of the following figure. Let XY, FZ, ZX be the intersections of the planes of acy, yz, »#, respectively, at the time #, with a sphere fixed in space, unity being its radius, and the origin its center ; then <a3 will be the resolved velocity along the great circle XY of any point P situated on that great circle, and o>j will be the resolved velocity along the great circle FZ of any point Q situated on that great circle, and o>2 will be the resolved velocity along the great circle ZX of any point R on that great circle; and these velocities tend in the directions represented by the arrows. 67 It is evident from this, that wj, o>2> «>3 are also the angular velocities of the planes of yz, ##, and ay, round the axes of #, y, and *, respectively : but it may be easily seen from the equations (.6), that wlt w25 <Ws are not tne angular velocities round the axes of #, y, ar, of any other points of the body, except those situated in the respective co-ordinate planes. 72. We may determine the position and motion of the body by means of these velocities, as follows. Let A be any fixed point on the surface of the sphere described before (see the figure), and AB any fixed great circle; draw the great circle AZC to meet YX produced in C. Let the angle ABZ = \^, the angle A Z = 0, and the angle C Z X = <p; then it is evident that these angles define completely the position of the body in space at the time t, and may be considered as the co-ordinates of the position of the body, \^ and 0 being the co-ordinates of the point Z, which we shall call the pole of the body, and (p the addi- tional co-ordinate requisite to determine the position of the plane of ZX, and therefore of the whole body; we may easily determine these co-ordinates from the quantities «!, w2, fc>3, as follows. It is evident that the velocities of the point of the body coinciding with Z are, , perpendicular to A Z, o/ 1 J f\ and — along ^Z, and the velocity along CX of the point of the body co- inciding with C is d\l/ --| sin AC, due to the variation of >Jr, + -T"5*0 ZC*» due to the variation of 0; 68 i. e., since ZC = 90°, Now by what we have proved in (71), the velocities of the body coinciding with Z are, &>2 along ZJT, and - wl along Z F, and the velocity along C X of the point coinciding with C is o>3 ; hence, since these two sets of velocities must be equivalent, we have, resolving the latter set so as to make them coincide with the former, -J- sin 9 = — MI cos 0 + o>2 sin < at — = ojj sin 0 + o)2 cos 0 (C). 73. These differential equations connect 0, \^, and 6 with «i, &)2» o>3, and these, along with the three others (A') in (69), which connect o^, w2, w3 with the impressed forces, form a system of six equations connecting the six unknown quantities o)15 w2, ft>3, \^, ^, 0 with #, and which therefore, when it is possible to do so, will enable us to solve any problem respecting the motion of a rigid body about its centre of gravity. The equations (B) will enable us to determine the mo- tion of any point we please of the body, should it be requisite to do so. 74. These equations will be sufficient for our present purpose, namely, the determination of the Earth's motion about its centre of gravity ; we shall hereafter recur to this subject, and deduce several interesting consequences from these equations. CHAPTER VI. PRECESSION AND NUTATION. 75. WE shall now make use of the equations deduced in the last Chapter, to determine the motion of the Earth round its center of gravity. The forces which act on the Earth, are the attractions of the Sun, Moon, and other planetary bodies; but on account of the Earth's nearly spherical form, the motions of the Earth round its center of gravity, produced by these forces, are but small ; hence, by the principle of the super- position of small motions, we may consider, separately and by itself, the effect of the attraction of each planetary body on the motion of the Earth round its center of gravity ; we shall accordingly commence with the Sun's effect. 76. We shall first prove, that the attractions of the Sun on any particle of the Earth are the same very nearly as if the Sun's mass were condensed into his center of gravity. Take the center of gravity of the Sun as origin; let xy'% be the co-ordinates of any element §m of it, and let xyz be the co-ordinates of the attracted particle §m of the Earth. 9 Then, if V denote the sum of each element of the Sun, divided by its distance from $m, the attractions of the Sun on $m will be dV dV dV dx" dy* d%' 70 \ Now V = 2 — 2 (o?^ + yy + #*') r' putting r and r' for <#2 + y2 4- #2, and X* 4- y'2 + #'2, re- spectively a?<j?' + yy + %% \ r } expanding and neglecting the squares, &c. of the very small x' 11 %' , r quantities — , — , — , and - r r r r 1 = — 2owi . r Since 2^m,t?', 25my', 2^m^' are zero, because the origin is center of gravity. Hence F= — ; r and therefore, neglecting the squares &c. of very small dV dV dV quantities, F, and consequently the attractions — , — , — , are the same as if the whole mass of the Sun were collected into its center of gravity. 77- Now take the principal axes of the Earth, passing through its center of gravity, as the axes of co-ordinates, the polar axis being that of % ; let aoy% be the co-ordinates of any element $m of the Earth, and xyss the co-ordi- nates of the Sun, supposed to be condensed into his center of gravity ; then, m being the mass of the Sun, his at- tractions on £m, parallel to the axes of x and y, will be andF= 71 and hence the quantity N, (see the equations (A') of the last Chapter) which equals 2$w (Yx — Xy)^ will become (y'-y)*-(*'-«>)y \ *'-*m^)2+(*'-*)i* '"I \yoe-aiy) [r'2-2(a!X+yy + **') + r2]"! \ , r and r being the distances of $m and $m from the origin, and r being therefore very large compared with a?, y\ %9 or r ; hence, expanding and neglecting the squares of very small quantities, we have or, since the origin is center of gravity, and therefore 2^ma?, S^my, 23m*, each zero, and since the axes are principal axes, and therefore each zero, we have and ^f being the same as in the last Chapter. Similarly, 8m' *V - Q, r Sm'y'ss' (C-B). r'5 Hence the equations (-4')j in the last Chapter, become A + (C - B) W2»3 = (C - K) 72 78. These equations simplify very much in the case of the Earth, for the polar axis being that of *, the moments of inertia round all axes in the plain of xy will be the same, since the Earth is symmetrical with respect to the polar axis ; hence B = A, and the last equation will become and therefore o>3 = constant = n suppose. That is to say, by (71), the angular velocity of the plane of the equator round the fixed axis with which the polar axis coincides at any time, is a constant quantity. Now if ri be the Sun's mean angular velocity relative to the Earth, n'2= — — , m being the Earth's mass, = — , very nearly, m being very small com- pared with TO'; C — A hence, and putting ft for — - — , the other two equations become 79- Now in the figure (page 66), let S be the point where a line drawn from the center of the Earth to the Sun meets the fixed sphere mentioned in page 66; draw the great circle SZ, then SZ will be the Sun's north polar dis- tance, which we shall denote by A, and SZX will be the Sun's hour-angle relative to the meridian plane ZX, which we shall denote by h : it is evident that A and h are the polar co-ordinates of the Sun, and therefore we have 73 x = r sin A cos A, y = r sin A sin A, z' = r cos A. hence -r- = sin A cos A sin A, ra — '— = sin A cos A cos h ; r a and hence our equations become — + w/3 wa = 3w'2/3 sin A cos A sin A, 2 - w/3o>i= — 3n'2fi sin A cos A cos A. ct£ We shall take a year as the unit of time, and hence n', 2ir which = - — , will become 27r, and n will be about 365, a year also we shall shew that /3 is about = — . 330 Now h varies in consequence of the Earth's diurnal rotation, and also in consequence of the Sun's motion and the motion of the polar axis ; but the part of its variation due to the former cause is very much greater than that due to the latter causes, hence we may put dh where $n is a small quantity compared with n, depending on the motions of the Sun and polar axis, we put - n because h decreases with the time. Hence in the first of our equations, putting d<a\ dh)\ dh dto, , * it becomes da,, nfi 3n'*fi e»>2 = — — s— sin A cos A sin n ; dh - n 4- $n - n + 10 74 or, neglecting $n compared with w, — - - /3ft>2= — Sra'* — sin A cos A sin h. ah n Now the second member of this equation is very much smaller than the first, on account of being multiplied by - , hence in it we may, when integrating, consider the n periodical quantity A as invariable, since it varies very slowly compared with h ; and therefore putting, for brevity, 3n'2ft . — sin A cos A = y, n our equation becomes ^ -£«,•• -y sin h... (1), where y is a very small quantity which may be considered invariable in integrating. In the same way we shall have -~- + ah 80. Now differentiating (l), and adding (2) multiplied by ft to it, we have _J2 -jj£ + ft2u,} = (- y + 7/3) cos h = - y cos h, neglecting ft compared with unity. The integral of this equation will be o>i = A cos (fih + B) + C cos h, where A and B are arbitrary constants, and C a constant to be determined by substitution. Now A and B, since they depend only on the initial circumstances of the motion, are independent of the Sun's action ; and, as it is our object to 75 determine the effect of that action alone, we shall omit the term Acos(fih+ B), which does not depend on it, and we shall have simply (Dl = C cos h, so far as the Sun's action is concerned ; substituting this value in the equation, we find C(-i + /r> = -7, or C = y, neglecting ft2 ; hence fcjj = y cos h. In like manner we shall have, differentiating the equa- tion (2), and subtracting (1) multiplied by /3 from it, d2a)2 „„ . — — + /52ft>2 = - 7 sin h ; an and therefore, as before, ft>2 = y sin h. 81. Having thus determined &>! and ft>25 we shall sub- stitute their values in the equations (C) (last Chapter), in order to determine 9 and 0, and so find the position and motion of the pole ; we have then, substituting the values of o>i and ft>2 just obtained in the two first of the equations (C), dvl/ — ~ sin 9 = y sin h sin 0 - y cos h cos <p = - y cos (h + 0), dO and — = y sin h cos (f> + y cos h sin 0 = y sin (h + 0). Now in the figure (page 66), the angle SZX is A, and the angle XZC is 0, hence the angle *S*ZC is (f) + h; there- fore if we take the point A to be the pole of the ecliptic (which we may do since its position is arbitrary), it is evi- dent that ^ + h or SZC will be the Sun's right ascension 76 — 90°; hence, if a be the Sun's right ascension, we shall have $ + h = a - 90°, and hence our equations become d^ . — — sin 9 = — y sin a, dO - = -7cosa; or, putting for y its value, d^ . 3n'2Q . — - sin 6 = — sin A cos A sin a, at n d9 3riz$ . — = sin A cos A cos a. dt n 82. Now if / be the Sun's longitude, /, a, and 90° - A are the sides of a right-angled spherical triangle, the right angle being opposite Z, and 0 (the obliquity of the ecliptic) being the angle opposite 90° — A ; hence since, by Napier's rules, cos A = sin 6 sin I (1), cos Z = sin A cos a (2), and sin a = cot A cot 9 (3), we have sin A cos A sin a = cos2 A cot 0, by (3), = sin 0 cos 9 sin2 Z, by (l), and sin A cos A cos a = cos Z cos A, by (2), = sin 9 sin I cos Z, by (l) ; hence our equations become <ty 3n'20 I_ = r cos 0 sm2 /, dt n d9 3n'*P . a- _ _ L- SJn 0 Cos /. dt n 77 83. Now — being very small, we may in integrating n these equations consider & invariable in the terms multiplied by — ; and we may also, for the same reason, consider I to n dl vary uniformly, and therefore put — = n ; hence, putting d\/ d\, dl d d9 dO , Tt=-din> and sin2 / = 1 (l - cos 2 Z), our equations become S— £* hence, integrating ^ + C = - - - y3 . cos 6 (I - J sin 2/), Q + C" = 3 - 3 sin 0 sin /. n ^ In the second members of these equations 0 may be considered as the mean obliquity of the ecliptic, and may therefore be determined by astronomical observations; and — , the ratio of a year to a day, may be similarly deter- n mined ; and thus we may put our equations in the form 9 + C = gfi sin /, where e', f ', g', are numerical quantities got from obser- vation. These equations determine the effect of the Sun's at- traction on the motion of the Earth round its center of gravity. 78 84. To determine the effect of the Moon on the Earth's motion round its center of gravity, we may proceed in ex- actly a similar way, merely supposing the symbols which before referred to the Sun now refer to the Moon, and alter- ing them accordingly, as follows : Let my, w/5 and rt be the mass, mean angular velocity, and distance of the Moon, then as in Art. (78), we shall have here m instead of being much smaller than m t as before, is much larger than it, and therefore cannot be neglected as before, let X = — , then n m , and therefore — ' = - — ; r, 3 X + 1 hence, in changing our equations so as to refer to the Moon, M* we must in substituting for n2 put — '— instead of nf for it. Hence, if we put dashes under the letters to denote that they refer to the Moon, we shall have for the Moon's effect on the Earth's motion round its center of gravity, s ' / - sin '"' Of course the point A in the figure, (page 66), is now supposed to be the pole of the Moon's orbit, and not the pole of the ecliptic. 79 85. In these expressions the coefficients are much smaller than before; so small that the periodical quantities multiplied by them, viz. sin 2ntt and sin nt^ which go through all their values in half a month, and a month respectively, may be neglected, this will give and and therefore --- = - - •—- — cos 9 , dt 2n (X + 1) , dB and — ' = 0, dt Which equations prove, that the effect of the Moon's attraction (omitting very small periodical quantities of short period) is to produce a motion of the pole of the Earth per- pendicular to the great circle AZ, i.e., the great circle joining the pole of the Earth and the pole of the Moon's orbit ; and the velocity with which this motion takes place (i.e. sin 0,), is cos 9 sin We shall resolve this velocity along and perpendicular to the great circle, joining the pole of the Earth and the pole of the ecliptic, in order to get our quantities measured in the same way as before in the case of the Sun, and so de- termine the variations (due to the Moon's action), of the angles \js and 0, which refer to the pole of the ecliptic. 86. Let i be the inclination of the Moon's orbit to the plane of the ecliptic ; then it is evident that t, 0, and 9t form the sides of a spherical triangle. 80 Let cr be the angle made by 6 and 0/? then resolving the velocity 3«2/3 — cos 9 sm 6 , 2w (X + 1) (which acts perpendicularly to 0y), along and perpendicular to 9 we find for the resolved parts, -- 7-* - - cos 9 sin 0 cos a perpendicular to 0, 2n (\ + 1) and -- -r-* - - cos 9f sin 9t sin cr along 0, 2 w (A. + 1} and hence, since — sin 9 and — are these velocities, we at at have the following equations to determine the effect of the Moon's action on \!/ and 9, viz. d\js 3nffl cos Qt sin 9f cos cr 1/7 2n(\+ 1) sin 0 d0 3w//3 — = -- — cos 9 sin 0 sin cr. dt Now in the triangle of which t, 0, and 0/ are the sides, o- is the angle opposite i, and if & be the longitude of the Moon's node, it is evident that & is the angle opposite Bt ; hence we have cos t = cos 9 cos 0/ + sin 0 sin 0y cos cr, ... (l) cos Qt = cos i cos 0 + sin i sin 0 cos & , ... (2) sin 0. sin t (3) am 56 alu O" (•n« rt sin A r-i\< rr hence sin0 COS t - COS 0 COS 0, = (cosi cosfl + siru sin0cos& ) 81 cos t sin2 0 — sin t cos0 sin 9 cos & sin2 9 (sin20-cos20) cos2i cos 0+sm t cos i — — — - - - cos & -sm2t cos 9 cos2 & sin 9 — : — — sin 9 and also cos 9, sin 9 1 sin a- = (cos i cos 9 + sin t sin 9 cos & ) sin t sin £ = ^ sin 2t cos 9 sin & + ^ sin8 1 sin 0 sin 2 & . In these expressions Q> alone may be considered as va- riable, all the other quantities varying very slowly, and within very small limits; also, since these expressions are to be multiplied by a very small coefficient, we may, in the periodical terms, neglect sin2t, since t is not much more than 5°. Also, if v be the mean angular retrograde velocity of the Moon's nodes, we may put and hence we shall have as in former cases 3w2/3 A sin2* 1 ' cos0{cos-< ----- sin 2 1 cot 29 cos Q }, i/(\+l) 1 2 j cos0sm2t sin & ; — - -- d& 2ni/(\ d9 and hence, integrating, 9 + C' = - - — / ^ ^ cos0 sin2t cos & ; 11 82 or, as before, we may put these equations in the form 3 3 ^tc-^ — «-/ A + 1 When e^ ft, and g are numerical quantities, got by ob- servation. Thus we have determined the effect of the Moon's attraction on the motion of the Earth, round its center of gravity. 87- The effects of the other planetary bodies are very small indeed, and we shall neglect them ; hence, adding together the effects of the Sun and Moon, we find for the whole motion of the Earth round its center of gravity, so far as it is affected by external attractions, >/, + C = - /3 (el - ~^~ Q ) + /'/3 sin 2 / A. + 1 0+ C' = 's\i / A + I The first term of the expression for >// + C is non- periodical, and its rate of variation is or -(3(e'n' + ^-_ which being constant and negative represents a uniform retrograde motion of the pole in longitude : it gives rise to what is called the precession of the equinoxes, because, in consequence of it, the first point of Aries moves constantly backwards, and therefore the equinox occurs sooner than it otherwise would every year. 83 Observation shews that this retrograde motion of the pole in longitude is about 50".l per year; hence we ought to have The other terras of \|/ + C and 0 + C' are periodical, depending on the longitude of the Sun and of the Moon's nodes; they are called the solar and lunar nutations. Ob- servation shews that the coefficient of sin & is about 18", and that of cos Q about 9".5, the coefficients of the other terms are much smaller ; hence we ought to have and 9».6-«r,r-- ............ (3). A + 1 88. Since e , //5 and g are known numerical quan- tities, it is evident, that from (l) combined with (2) or (3), we may eliminate )3 and find X ; the result is X = about 70, thus by observation on precession and nutation we may determine X, which is the ratio of the Earth's mass to that of the Moon. By the same equations we may determine /3, the re- sult is /3 = about .00319, or - . *oO 89- Now /3 may be also calculated by integration, if we know the law of arrangement of the Earth's mass ; we shall calculate /3, assuming the results arrived at in Chap, in, and if we find that the value of /3 thus obtained coincides with that just determined by observation, it is evident that we shall have an additional proof of the correctness of our hypotheses in Chap. in. 84 90. We have evidently »-») 2Sm (*»-«») ' S1 #, y, # being the co-ordinates of any particle ($m) of the Earth, the polar axis being the axis of ss ; or using the polar co-ordinates as before, Now putting ^ + ^ cos 20 for cos20, and integrating relatively to 0, observing that r does not contain 0, the numerator of )3 becomes which, putting r'=«5{l +5e(J -!.")}, and observing the property of Laplace^s coefficients in Art. 21, becomes 9 9 - 8 ~ 45' hence our integral becomes Sir ra d(o5e) . — / p -- da. 15 Jn da 85 In like manner the denominator of /3 becomes TT f r«, d(O/4 /i \1 = - / / p— : — { -- -- M V 5 J-\ Jo r da (3 \3 ) I Since, on account of the smallness of /3, we may neglect e in its denominator. Hence, observing the property in Art. 21, the denomi nator of /3 becomes STT /•<•! rf(a5) STT /•«. — / p — — aa which = — / pa*da. 15 Jo * da 3 A ^ hence we have 5 / p 1 1 <J Q (t (t r^ I pa*da «/o Now the equation for determining e in Art. 53, gives e, /•"' . 1 ra' d(a5e) a>zaj -- ^ / pa'da + — - / p— — da + -- ^ = 0, a,J0 r *«/•'• ^a STT r"> w^o/ a e / pa^da -- '- 0 ' Vt r STT hence p = - rpa* da . ' a~ I pa? da ' Jo ' I pa*da putting m being, as before, the ratio of the centrifugal force to gravity at the equator. We can go no farther in calculating /3 without knowing the law of density ; hence, taking the law already assumed, and substituting in the integrals and performing the integrations, we shall find /3 = about .0031359, or . 330 91. 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BABINGTON, M.A., Fellow of St. John's College, Cambridge. 8vo. 4s. 6d. CAMBRIDGE UNIVERSITY ALMANAC for the Year 1852. Embellished with a Line Engraving, by Mr. E. CHALLIS, of a View of the Interior of Trinity College Library, from a Drawing by B. RUDGE. (Con- tinued Annually.) 4s. 6d. DESCRIPTIONS of the BRITISH PALEOZOIC FOSSILS added by Professor Sedgwick to the Woodwardian Collection, and contained in the Geological Museum of the University of Cambridge. With Figures of the new and imperfectly known Species. By F. M'COY, Professor of Geology, &c., Queens College, Belfast ; Author of " Car- boniferous Limestone Fossils of Ireland," '• Synopsis of the Silurian Fossils of Ireland." Parti. 4to. GRADUATI CANTABRIGIENSES : sive Cata- logus eorum quos ab anno 1760 usque ad 10m Octr. 1846, Gradu quocunque ornavit Academia. Cura J. ROMLLLY, A.M., Coll. Trin. Socii atque Academise Registrarii. 8vo. 10s. MAKAMAT; or Rhetorical Anecdotes of Hariri of Basra, translated from the Arabic into English Prose and Verse, and illustrated with Annotations. By THEODORE PRESTON, M.A., Fellow of Trinity College, Cambridge. Royal 8vo. 18s.; large paper, II. 4s. Cambridge. 24 Miscellaneous Works. ARCHITECTURAL NOTES ON GERMAN CHURCHES ; with Notes written during an Architectural Tour in Picardy and Normandy. By W. WHEWELL, D.D., Master of Trinity College, Cambridge. Third Edition. To which is added, Translation of Notes on Churches of the Rhine, by M. F. De LASSATJLX, Architectural Inspector to the King of Prussia. Plates. 8vo. 12s. REMARKS on the ARCHITECTURE of the MIDDLE AGES, especially in Italy. By R.WILLIS, M.A., Jacksonian Professor of the University of Cambridge. Plates, LAEGE PAPER. Royal 8vo. il. 1*. Views of the Colleges and other Public Buildings IN THE UNIVERSITY OF CAMBRIDGE, Taken expressly for the UNIVERSITY ALMANACK, (measuring abo-ut 17 inches by 11 inches). Cambridge Antiquarian Society's Publications, Nos. I. to XV. Demy 4to., sewed. A 000426824 9 Sh'n'tlii n-ilt hs PuUixlicd. !>;! the wwr Aitthnr : MATHEMATICAL TRACTS, PART II. Containing, among other subjects, The MATHEMATICAL THEORY of ELECTRICITY, briefly considered,— The LIBRATIONS of the MOON, ^- The THEORY of JUPITER'S SATELLITES; and a Short ESSAY on TIDES.